9x^2 + 4y^2 - 36x + 16y + 36 = 0

["Understanding the Conic Section: The Equation 9x² + 4y² - 36x + 16y + 36 = 0", "When analyzing the second-degree equation 9x² + 4y² - 36x + 16y + 36 = 0, it reveals a conic section — specifically an ellipse — hidden within a seemingly standard quadratic form. While the equation contains both x² and y² terms with positive coefficients, their differing coefficients suggest an elongated ellipse rather than a circle. In this article, we explore how to rewrite the equation in standard ellipse form, interpret its geometric properties, and understand its significance in geometry and applications.", "---", "### Step 1: Rewrite and Complete the Square", "Start with the original equation:", "$$\n9x^2 + 4y^2 - 36x + 16y + 36 = 0\n$$", "Group x- and y-terms:", "$$\n(9x^2 - 36x) + (4y^2 + 16y) + 36 = 0\n$$", "Factor out coefficients of squared terms:", "$$\n9(x^2 - 4x) + 4(y^2 + 4y) + 36 = 0\n$$", "Now complete the square inside each group:", "- For x: $x^2 - 4x$, take half of –4 (which is –2), square it: $(–2)^2 = 4$\n Add and subtract $4 \cdot 9 = 36$", "- For y: $y^2 + 4y$, take half of 4 (which is 2), square it: $2^2 = 4$\n Add and subtract $4 \cdot 4 = 16$", "Rewriting:", "$$\n9[(x^2 - 4x + 4) - 4] + 4[(y^2 + 4y + 4) - 4] + 36 = 0\n$$", "$$\n9(x - 2)^2 - 36 + 4(y + 2)^2 - 16 + 36 = 0\n$$", "Combine constants:", "$$\n9(x - 2)^2 + 4(y + 2)^2 - 16 = 0\n$$", "Move constant to the right:", "$$\n9(x - 2)^2 + 4(y + 2)^2 = 16\n$$", "---", "### Step 2: Standard Form of an Ellipse", "Divide every term by 16 to achieve 1 on the right-hand side:", "$$\n\frac{9(x - 2)^2}{16} + \frac{4(y + 2)^2}{16} = 1\n$$", "Simplify:", "$$\n\frac{(x - 2)^2}{\frac{16}{9}} + \frac{(y + 2)^2}{4} = 1\n$$", "$$\n\frac{(x - 2)^2}{\left(\frac{4}{3}\right)^2} + \frac{(y + 2)^2}{2^2} = 1\n$$", "This is the standard form of an ellipse:", "$$\n\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\n$$", "Where:\n- Center: $(h, k) = (2, -2)$\n- Semi-major axis length: $b = 2$ (since $b^2 = 4 > a^2 = \left(\frac{4}{3}\right)^2$)\n- Semi-minor axis length: $a = \frac{4}{3}$", "Because $b > a$, the major axis is vertical.", "---", "### Step 3: Key Properties", "- Center: The ellipse is centered at $(2, -2)$\n- Vertices:\n • Along major axis (y-direction): $(2, -2 \pm 2)$ → $(2, 0)$ and $(2, -4)$\n • Along minor axis (x-direction): $(2 \pm \frac{4}{3}, -2)$ → $\left(\frac{10}{3}, -2\right)$ and $\left(\frac{2}{3}, -2\right)$\n- Foci: Calculated using $c = \sqrt{b^2 - a^2} = \sqrt{4 - \frac{16}{9}} = \sqrt{\frac{20}{9}} = \frac{2\sqrt{5}}{3}$\n Foci located along the major axis at $(2, -2 \pm c) = \left(2, -2 \pm \frac{2\sqrt{5}}{3}\right)$", "---", "### Step 4: Geometric Interpretation", "This ellipse represents all points whose weighted distance sum from two foci is constant. Its elongated shape—vertical major axis—reflects the dominance of the larger denominator under $y$. This form is useful in physics, engineering, and computer graphics for modeling elliptical paths, optimizing shapes, and designing reflective surfaces.", "---", "### Step 5: Applications and Connections", "Equations of ellipses like this appear in:", "- Orbital mechanics, where planetary orbits follow elliptical paths\n- Acoustic engineering, in designing elliptical rooms or whispering galleries\n- Computer vision, for shape recognition and filtering\n- Graphics design, generating smooth curved forms", "---", "### Conclusion", "The original equation $9x^2 + 4y^2 - 36x + 16y + 36 = 0$, while algebraically quadratic, unveils a clean, symmetric ellipse centered at $(2, -2)$, elongated vertically. By completing the square and normalizing, we uncover essential geometric features—center, axes, vertices, foci—essential for visualization and practical use in science and technology. Understanding conic sections in this way transforms abstract algebra into tangible shape design.", "---", "### Frequently Asked Questions (FAQ)", "Q: Is this equation a circle?\nA: No, because the coefficients of $x^2$ and $y^2$ differ (9 vs. 4), indicating an elongated ellipse.", "Q: How do I sketch this ellipse?\nA: Plot the center at $(2, -2)$. Mark major axis along vertical line $x = 2$ from $(2, -4)$ to $(2, 0)$. Mark minor axis equally on horizontal $y = -2$ from $(2/3, -2)$ to $(10/3, -2)$. Draw smoothly joining these points in vertical orientation.", "Q: Can this be used in real-world modeling?\nA: Yes, elliptical equations are foundational in modeling orbits, astronomical data, automotive headlights, and architectural arches.", "---", "Optimize your understanding of conic sections today — from equations to ellipses to engineering!"]